© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

4.3.4.3 Nanopendulum Tachometry

The smallest angular velocity detectable using a simple nanopendulum is given approximately by w_min ~ Dx / r t_meas = 2 milliradian/sec (~0.1 deg/sec) for Dx = 1 nm, r = 500 nm, and t_meas = 1 sec. The largest detectable angular velocity w_max ~ 2 p / t_meas if t_meas is the minimum time required to count one complete rotation of the sensor; the smallest t_meas ~ 2 p / w_max ~ 100 nanosec for w_max ~ 49 megaradians/secor ~5% of the bursting speed (Eqn. 4.17) for an r = 1 micron spherical diamondoid nanodevice.

What rotational displacements will normally be encountered by an in vivo medical nanorobot? From Eqn. 3.2, a 1-micron device experiences instantaneous Brownian rotations of w_B = (kT / 4 p h R³ t)^1/2 radians/sec, where t = (MW_kg / 32 p kT N_A c_H2O R⁴)^1/2, the mean time between molecular collisions with the device (mostly by water molecules), from Eqn. 4.3, where c_H2O ~ 3.3 x 10²⁸ molecules/m³. For R = 0.5 microns and T = 310 K, t ~ 6 x 10^-20 sec/collision and w_B ~ 7 x 10⁹ rad/sec, with each displacement of ~ t w_B = 4 x 10^-10 radian, none of which are detectable. However, the net of all Brownian rotational displacements is one rotation every 16 sec, ~0.4 radian/sec (Section 3.2.1).

The largest measurable angular displacements in vivo are caused by tumbling due to differential shear forces in the bloodstream. Normal vessel wall shear rates in physiological bloodflow range from 100-1400 radians/sec in the larger arteries to 500-4000 radians/sec in the smaller arteries and capillaries.³⁸⁶ The measured rolling velocity of white cells during vascular "tank tread" locomotion (Section 9.4.3.6) on venule walls is ~40 micron/sec, or ~8 radians/sec, which may be encountered by nanodevices employing cytocarriage (Section 9.4.7). The human vestibular mechanism has a frequency response range of 0.048-260 radians/sec.⁴⁴⁹

Thus, the maximum range of physiologically relevant rotation rates is 10^-2 to 10⁴ radians/sec. Since the natural harmonic frequency of a simple nanopendulum is w_res = (g/r)^1/2 = 4400 radians/sec for r = 500 nm, the possibility of forced oscillations and resonance need be faced only when measuring the highest likely physiological rotation rates, in which case the addition of a damping force F_damp ~ m v_turn w_res during resonance events should eliminate the problem. F_damp for critical damping peaks at ~mg ~ 0.01 pN for the m = 10^-15 kg, r = 500 nm case (v_turn ~ 2 mm/sec).

Last updated on 17 February 2003